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18
Evaluation of Collision Detection Methods for Virtual Reality FlyThroughs
 In Canadian Conference on Computational Geometry
, 1995
"... We consider the problem of preprocessing a scene of polyhedral models in order to perform collision detection very efficiently for an object that moves amongst obstacles. This problem is of central importance in virtual reality applications, where it is necessary to check for collisions at realtime ..."
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Cited by 71 (7 self)
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We consider the problem of preprocessing a scene of polyhedral models in order to perform collision detection very efficiently for an object that moves amongst obstacles. This problem is of central importance in virtual reality applications, where it is necessary to check for collisions at realtime rates. We give an algorithm for collision detection that is based on the use of a mesh (tetrahedralization) of the free space that has (hopefully) low stabbing number. The algorithm has been implemented and tested, and we give experimental results comparing its performance against three other algorithms that we implemented, based on standard data structures. A preliminary version of this paper appeared in the proceedings of the 7 th Canad. Conf. Computat. Geometry, Qu'ebec, Aug 1013, 1995. y held@ams.sunysb.edu; Supported by NSF Grant DMS9312098. On sabbatical leave from Universitat Salzburg, Salzburg, Austria. z jklosow@ams.sunysb.edu; Supported by NSF grants ECSE8857642 and C...
Separation and Approximation of Polyhedral Objects
, 1993
"... Given a family of disjoint polygons P1, P2, : ::, Pk in the plane, and an integer parameter m, it is NPcomplete to decide if the Pi's can be pairwise separated by a polygonal family with at most m edges, that is, if there exist polygons R1; R2; : ::; Rk with pairwisedisjoint boundaries such t ..."
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Cited by 29 (3 self)
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Given a family of disjoint polygons P1, P2, : ::, Pk in the plane, and an integer parameter m, it is NPcomplete to decide if the Pi's can be pairwise separated by a polygonal family with at most m edges, that is, if there exist polygons R1; R2; : ::; Rk with pairwisedisjoint boundaries such that Pi Ri andP jRij m. In three dimensions, the problem is NPcomplete even for two nested convex polyhedra. Many other extensions and generalizations of the polyhedral separation problem, either to families of polyhedra or to higher dimensions, are also intractable. In this paper, we present e cient approximation algorithms for constructing separating families of nearoptimal size. Our main results are as follows. In two dimensions, we give an O(n log n) time algorithm for constructing a separating family whose size is within a constant factor of an optimal separating family; n is the number of edges in the input family of polygons. In three dimensions, we show how to separate a convex polyhedron from a nonconvex polyhedron with a polyhedral surface whose facetcomplexity is O(log n) times the optimal, where n = jPj+jQj is the complexity of the input polyhedra. Our algorithm runs in O(n4) time, but improves to O(n3) time if the two polyhedra are nested and convex. Our algorithm for separating a convex polyhedron from a nonconvex polyhedron extends to higher dimensions. In d dimensions, for d 4, the facetcomplexity of the approximation polyhedron is O(d log n) times the optimal, and the algorithm runs in O(nd+1) time. Finally, we also obtain results on separating sets of points, a family of convex polyhedra, and separation by nonpolyhedral surfaces, such as spherical patches.
On the Complexity of Optimization Problems for 3Dimensional Convex Polyhedra and Decision Trees
 Comput. Geom. Theory Appl
, 1995
"... We show that several wellknown optimization problems involving 3dimensional convex polyhedra and decision trees are NPhard or NPcomplete. One of the techniques we employ is a lineartime method for realizing a planar 3connected triangulation as a convex polyhedron, which may be of independent i ..."
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Cited by 19 (0 self)
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We show that several wellknown optimization problems involving 3dimensional convex polyhedra and decision trees are NPhard or NPcomplete. One of the techniques we employ is a lineartime method for realizing a planar 3connected triangulation as a convex polyhedron, which may be of independent interest. Key words: Convex polyhedra, approximation, Steinitz's theorem, planar graphs, art gallery theorems, decision trees. 1 Introduction Convex polyhedra are fundamental geometric structures (e.g., see [20]). They are the product of convex hull algorithms, and are key components for problems in robot motion planning and computeraided geometric design. Moreover, due to a beautiful theorem of Steinitz [20, 38], they provide a strong link between computational geometry and graph theory, for Steinitz shows that a graph forms the edge structure of a convex polyhedra if and only if it is planar and 3connected. Unfortunately, algorithmic problems dealing with 3dimensional convex polyhedra ...
Efficient Contact Determination Between Geometric Models
 INTERNATIONAL JOURNAL OF COMPUTATIONAL GEOMETRY AND APPLICATIONS
"... The problem of interference detection or contact determination between two or more objects in dynamic environments is fundamental in computer graphics, robotics and computer simulated environments. Most of the earlier work is restricted to either polyhedral models or static environments. In this pap ..."
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Cited by 11 (3 self)
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The problem of interference detection or contact determination between two or more objects in dynamic environments is fundamental in computer graphics, robotics and computer simulated environments. Most of the earlier work is restricted to either polyhedral models or static environments. In this paper, we present efficient algorithms for contact determination and interference detection between geometric models undergoing rigid motion. The set of models include polyhedra and surfaces described by algebraic sets or piecewise algebraic functions. The algorithms make use of temporal and spatial coherence between successive instances and their running time is a function of the motion between successive instances. The main characteristics of these algorithms are their simplicity and efficiency. They have been implemented; their performance on many applications indicates their potential for realtime simulations.
Efficient Collision Detection for Interactive 3D Graphics and Virtual Environments
, 1998
"... Collision detection is of paramount importance for many applications in computer graphics and visualization. Typically, the input to a collision detection algorithm is a large number of geometric objects comprising an environment, together with a set of objects moving within the environment. In add ..."
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Cited by 11 (0 self)
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Collision detection is of paramount importance for many applications in computer graphics and visualization. Typically, the input to a collision detection algorithm is a large number of geometric objects comprising an environment, together with a set of objects moving within the environment. In addition to determining accurately the contacts that occur between pairs of objects, one needs also to do so at realtime rates. Applications such as haptic forcefeedback can require over 1000 collision queries per second. We initially analyze several methods of performing collision detection which utilize standard boxbased data structures. These methods are compared iii against a more sophisticated technique, based upon recent computational geomet...
Generalized penetration depth computation
 In SPM ’06: Proceedings of the 2006 ACM symposium on Solid and physical modeling
, 2006
"... Penetration depth (PD) is a distance metric that is used to describe the extent of overlap between two intersecting objects. Most of the prior work in PD computation has been restricted to translational PD, which is defined as the minimal translational motion that one of the overlapping objects must ..."
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Cited by 11 (2 self)
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Penetration depth (PD) is a distance metric that is used to describe the extent of overlap between two intersecting objects. Most of the prior work in PD computation has been restricted to translational PD, which is defined as the minimal translational motion that one of the overlapping objects must undergo in order to make the two objects disjoint. In this paper, we extend the notion of PD to take into account both translational and rotational motion to separate the intersecting objects, namely generalized PD. When an object undergoes rigid transformation, some point on the object traces the longest trajectory. The generalized PD between two overlapping objects is defined as the minimum of the longest trajectories of one object under all possible rigid transformations to separate the overlapping objects. We present three new results to compute generalized PD between polyhedral models. First, we show that for two overlapping convex polytopes, the generalized PD is same as the translational PD. Second, when the complement of one of the objects is convex, we pose the generalized PD computation as a variant of the convex containment problem and compute an upper bound using optimization techniques. Finally, when both the objects are nonconvex, we treat them as a combination of the above two cases, and present an algorithm that computes a lower and an upper bound on generalized PD. We highlight the performance of our algorithms on different models that undergo rigid motion in the 6dimensional configuration space. Moreover, we utilize our algorithm for complete motion planning of polygonal robots undergoing translational and rotational motion in a plane. In particular, we use generalized PD computation for checking path nonexistence.
Computing Largest Circles Separating Two Sets of Segments
, 1996
"... A circle C separates two planar sets if it encloses one of the sets and its open interior disk does not meet the other set. A separating circle is a largest one if it cannot be locally increased while still separating the two given sets. An \Theta(n log n) optimal algorithm is proposed to find all l ..."
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Cited by 10 (0 self)
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A circle C separates two planar sets if it encloses one of the sets and its open interior disk does not meet the other set. A separating circle is a largest one if it cannot be locally increased while still separating the two given sets. An \Theta(n log n) optimal algorithm is proposed to find all largest circles separating two given sets of line segments when line segments are allowed to meet only at their endpoints. In the general case, when line segments may intersect\Omega\Gamma n 2 ) times, our algorithm can be adapted to work in O(nff(n) log n) time and O(nff(n)) space, where ff(n) represents the extremely slowly growing inverse of the Ackermann function.
Circular Separability of Polygons
, 1994
"... Two planar sets are circularly separable if there exists a circle enclosing one of the set and whose open interior disk does not intersect the other set. This paper studies two problems related to circular separability. A lineartime algorithm is proposed to decide if two polygons are circularly sep ..."
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Cited by 7 (2 self)
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Two planar sets are circularly separable if there exists a circle enclosing one of the set and whose open interior disk does not intersect the other set. This paper studies two problems related to circular separability. A lineartime algorithm is proposed to decide if two polygons are circularly separable. The algorithm outputs the smallest separating circle. The second problem asks for the largest circle included in a preprocessed, convex polygon, under some point and#or line constraints. The resulting circle must contain the query points and it must lie in the halfplanes delimited by the query lines.
Geometric intersection
 Handbook of Discrete and Computational Geometry, chapter 33
, 1997
"... Detecting whether two geometric objects intersect and computing the region of intersection are fundamental problems in computational geometry. Geometric intersection problems arise naturally in a number of applications. Examples include geometric packing and covering, wire and component layout in VL ..."
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Cited by 6 (0 self)
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Detecting whether two geometric objects intersect and computing the region of intersection are fundamental problems in computational geometry. Geometric intersection problems arise naturally in a number of applications. Examples include geometric packing and covering, wire and component layout in VLSI, map overlay
DETERMINING THE SEPARATION OF SIMPLE POLYGONS
 INTERNATIONAL JOURNAL OF COMPUTATIONAL GEOMETRY & APPLICATIONS
, 1993
"... Given simple polygons P and Q, their separation, denoted by (P; Q), is de ned to be the minimum distance between their boundaries. We present a parallel algorithm for finding a closest pair among all pairs (p; q), p 2 P and q 2 Q. The algorithm runs in O(log n) time using O(n) processors on a CREW P ..."
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Cited by 4 (0 self)
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Given simple polygons P and Q, their separation, denoted by (P; Q), is de ned to be the minimum distance between their boundaries. We present a parallel algorithm for finding a closest pair among all pairs (p; q), p 2 P and q 2 Q. The algorithm runs in O(log n) time using O(n) processors on a CREW PRAM, where n = jP j + jQj. This algorithm is timeoptimal and improves by a factor of O(log n) on the time complexity of previous parallel methods. The algorithm can be implemented serially in (n) time, which gives the first optimal sequential algorithm for determining the separation of simple polygons. Our results are obtained by providing a unified treatment of the separation and the closest visible vertex problems for simple polygons.